Magnetic Vector Potential Formulation Research Articles

Structure-preserving and efficient numerical simulation for diffuse interface model of two-phase magnetohydrodynamics

In this paper, we propose a novel diffuse interface model of two-phase magnetohydrodynamics (MHD) based on a magnetic vector potential formulation in the three-dimensional case. This model ensures an exact divergence-free approximation of the magnetic field by introducing a magnetic vector potential A and defining the magnetic field by B=curlA. The resulting framework constitutes a highly coupled, nonlinear saddle point system consisting of the Cahn–Hilliard system and MHD potential system. To solve the model efficiently, we present two fully decoupled, first-order, linear, and unconditionally energy-stable schemes and strictly prove their well-posedness and energy stability. Finally, we present several numerical examples that demonstrate the stability and effectiveness of our schemes.

Efficient Reduced Magnetic Vector Potential Formulation for the Magnetic Field Simulation of Accelerator Magnets

Efficient Reduced Magnetic Vector Potential Formulation for the Magnetic Field Simulation of Accelerator Magnets

Error analysis of the linearized Crank-Nicolson FEM for the incompressible vector potential magnetohydrodynamic system

A linearized fully discrete Crank-Nicolson finite element scheme is proposed for solving the three-dimensional incompressible magnetohydrodynamic equations based on a magnetic vector potential formulation, where the magnetic induction is written to a rotation of magnetic vector potential. By using the MINI element and lowest order Nédélec edge element to approximate the velocity field and pressure of fluid and magnetic vector potential, respectively, the numerical solution of magnetic induction can preserve the exactly divergence-free condition in fully discrete level. Error estimates for the velocity field and magnetic vector potential are rigorously analyzed under some reasonable regularity assumptions of exact solution. Finally, numerical results are given to support the theoretical analysis.

Two-dimensional Axisymmetric Modeling of the Magnetic Field for High-voltage Gas Circuit Breakers

High fidelity numerical analyses of high-voltage gas circuit breakers have been conducted at Hitachi Energy Research with an in-house CFD-based arc simulation tool. The tool extends the capability of a commercial flow solver (ANSYS Fluent) to represent physical phenomena at play in a high voltage circuit breaker during a breaking operation, such as magnetostatics, polymeric and metal evaporation, and arc-network interaction. This work describes the implementation of the Biot-Savart law for computing the magnetic field generated by an electric arc under the magnetostatic approximation and in two--dimensional axisymmetric conditions. The implementation is compared to the reference one based on the magnetic vector potential formulation of the Amp`ere’s law in the Coulomb gauge. The limitations of the two formulations are discussed and their numerical accuracy compared.

A hybrid enrichment strategy for the efficient construction of reduced order models by the proper generalized decomposition

PurposeThe calculation of electromagnetic fields can involve many degrees of freedom (DOFs) to achieve accurate results. The DOFs are directly related to the computational effort of the simulation. The effort is decreased by using the proper generalized decomposition (PGD) and proper orthogonalized decomposition (POD). The purpose of this study is to combine the advantages of both methods. Therefore, a hybrid enrichment strategy is proposed and applied to different electromagnetic formulations.Design/methodology/approachThe POD is an a-priori method, which exploits the solution space by decomposing reference solutions of the field problem. The disadvantage of this method is given by the unknown number of solutions necessary to reconstruct an accurate field representation. The PGD is an a-priori approach, which does not rely on reference solutions, but require much more computational effort than the POD. A hybrid enrichment strategy is proposed, based on building a small POD model and using it as a starting point of the PGD enrichment process.FindingsThe hybrid enrichment process is able to accurately approximate the reference system with a smaller computational effort compared to POD and PGD models. The hybrid enrichment process can be combined with the magneto-dynamic T-Ω formulation and the magnetic vector potential formulation to solve eddy current or non-linear problems.Originality/valueThe PGD enrichment process is improved by exploiting a POD. A linear eddy current problem and a non-linear electrical machine simulation are analyzed in terms of accuracy and computational effort. Further the PGD-AV formulation is derived and compared to the PGD-T-Ω reduced order model.

Analysis of the convergence of Newton method by finite element simulation with vector hysteresis stop model

PurposeThe goal of this research is to investigate the convergence behavior of the Newton iteration, when solving the nonlinear problem with consideration of hysteresis effects. Incorporating the vector hysteresis model in the magnetic vector potential formulation has encountered difficulties. One of the reasons is that the Newton method is very sensitive regarding the starting point and states distinct requirements for the nonlinear function in terms of monotony and smoothness. The other reason is that the differential reluctivity tensor of the material model is discontinuous due to the properties of the stop operators. In this work, line search methods to overcome these difficulties are discussed.Design/methodology/approachTo stabilize the Newton iteration, line search methods are studied. The first method computes an error-oriented search direction. The second method is based on the Wolfe-Powell rule using the Armijo condition and curvature condition.FindingsIn this paper, the differentiation of the vector stop model, used to evaluate the Jacobian matrix, is studied. Different methods are applied for this nonlinear problem to ensure reliable and stable finite element simulations with consideration of vector hysteresis effects.Originality/valueIn this paper, two different line search Newton methods are applied to solve the magnetic field problems with consideration of vector hysteresis effects and ensure a stable convergence successfully. A comparison of these two methods in terms of robustness and efficiency is presented.

Numerical study on flux-jump occurrence in a cup-shaped MgB2 bulk for magnetic shielding applications

MgB2 is one of the most promising materials for superconducting bulk applications. However, thermomagnetic instabilities can arise in the material because of its low heat capacity and thermal conductivity as well as its high critical current density. Being able to predict these phenomena, can guide and optimize MgB2-based devices for magnetic flux shielding or trapping applications. In this work, the flux-jump occurrence in an MgB2 cup-shaped shield is numerically studied using the finite element method by means of the commercial software COMSOL 6.0 Multiphysics®. To this aim, we developed a 2D axial-symmetric model coupling the heat diffusion equation and the magnetic equations based on a magnetic vector-potential () formulation. The comparison of the computed shielding curves with the experimental ones evidenced a good agreement between the two sets of data at different temperatures and positions along the shield’s axis. The as-validated model was then exploited to investigate possible optimization routes via the improvement of both the thermal conductivity of the material and the thermal exchange between the device and the cooling stage.

Parametric study for the induced electric current of an electromagnetically driven flow

In this work, the electric scalar and the magnetic vector potential magnetohydrodynamic formulations are implemented in the COMSOL Multiphysics software for the simulation of an electrovortex flow in a cuboid vessel. The flow is generated by a Lorentz force produced by the interaction of a direct electric current, which is injected through two solid electrodes located on the top and bottom of the vessel and an external magnetic field generated by one or a pair of permanent magnets. A good comparison between numerical and experimental velocity profiles reported by the workgroup is observed. The induced electric current fields obtained from Ohm's and Ampere's laws are explored in the solid electrodes and liquid metal regions. A parametric study for the interaction parameter in the range of 1 ≤ N ≤ 50 is presented. For both formulations, the same induced electric current distributions were obtained. Interestingly, for a pair of magnets, the induced electric current distribution is symmetric and the penetration of its main component into the solid electrodes increases with N, whereas for one single magnet, the induced current is non-symmetric and its penetration is small. Results also indicate that depending on the MF there are zones, where the induced electric current is small and the velocity is high. For a pair of magnets, this behavior is observed close to the interface of the LM-solid electrodes. Tables 2, Figs 10, Refs 43.

Study of Electromagnetic Shielding for Reduction of Magnetic Fields Generate by Underground Power Cables via a Finite Element Techinique

Nowadays, with the development of the industries, the electrical types of equipment are used and applied widely for many different purposes. One of them is the use of power cables to transmit electricity over long distances, from generator stations to consumers. Local and global fields are usually created by the extensive presence of the types of currents following in power cables in general and underground cables in particular. Not only causes directly some unwanted effects on the body’s natural equilibrium, but electromagnetic fields can create the disturbance and cause interference with electronic devices. In this paper, a shielding plate is introduced to mitigate the local fields generated by a three phase current of underground power cables. The computation of magnetic fields is implemented with the magnetic vector potential formulations via a finite element technique. The developed method is applied to a practical problem.

An Investigation of Magnetic Field Influence in Underground High Voltage Cable Shields

Magnetic fields and the shielding efficiency of the shields of underground high voltage cables are studied in this paper regarding several shielding configurations and materials. Shielding efficiency and magnetic fields are computed for shields with the same mesh but from different shielding materials, such as aluminum, ferrite, metal, and steel. In order to get the best shield configuration depending on the source characteristics and the material, a conducting ferromagnetic region with various thickness values is considered as shielding. A finite element model is introduced to investigate the influence of the parameters of magnetic fields and the shielding efficiency of underground high voltage cables. Furthermore, the reduction of the magnetic fields with or without shieldings is also presented. The developed method is performed with the magnetic vector potential formulations and validated on a practical problem.

Hybrid A-EHS Method for Fast Computation in Superconducting Bulks

In this article, a two-dimensional hybrid finite-element formulation is proposed to further reduce the computational time required for the time simulation of superconducting (SC) bulks. Low computational time formulations are important for the design stage of electrical machines with superconductors, where optimization tools are typically used. The proposed hybrid formulation is based on the partitioned <i>E</i>-<i>H</i> electromagnetic scheme (<i>EHS</i>) formulation and the magnetic vector potential formulation (<i>A</i>-formulation). The <i>EHS</i> formulation is used to solve the domains with highly nonlinear electric materials, while the <i>A</i>-formulation is used to solve the nonelectric domains. The main characteristics of the <i>EHS</i> are its partitioned formulation, where the matrix form of Amp&#x00E8;re&#x0027;s and Faraday&#x0027;s laws is separated and computed <i>a priori</i>, and the electric resistivity is defined at each node. The <i>EHS</i> formulation is first compared with the <i>H</i>-formulation for a single SC domain and then integrated with an <i>A</i>-formulation and compared with the <i>H</i> and <i>H</i>-<i>&phi;</i> formulations to simulate a bulk surrounded by air. Results verify the excellent accuracy of the <i>EHS</i> and <i>A-EHS</i> formulations, and a promising reduction of computational time is observed, with more emphasis on low levels of magnetization of the SC sample. When compared with the <i>H</i>-<i>&phi;</i> formulation, a time reduction between 48&#x0025; (6 T) and 90&#x0025; (1 T) is obtained.

Analysis of Leakage Inductances in Shunt Reactors: Application to High Voltage Transmission Lines

Inductance is one of the main parameters directly related to the reactive powers of Shunt Reactors (SRs) in electrical systems. Thus, the definition and computation of leakage inductances and the ratio of leakage to total inductances play an extremely important role in the design and manufacturing of SRs. In this study, a finite element approach was developed to compute leakage and total inductances and define a relationship between them with different SR powers and high voltage levels. The expanded method is presented with the magnetic vector potential formulations.

Numerical simulation methods for motion-induced eddy current testing signals based on Ar formulation and edge finite elements

In this paper, numerical methods for simulation of the motion-induced eddy current testing (MIECT) signals were developed based on the reduced magnetic vector potential formulation (Ar method) and the edge finite element. According to the principle of the MIECT problem, numerical methods with reference frame referring to the transducer or the inspection object were developed respectively to solve the eddy current and pickup magnetic field signals. In addition, a quasi-static numerical scheme was also adopted to simulate the MIECT problem for inspecting crack along the moving direction. Based on the proposed numerical schemes and the conventional Ar code, numerical programs of the 3 methods were developed and validated for simulating MIECT signals. Through comparing the numerical results and efficiencies of the three simulation schemes each other, the validity of the proposed methods and the corresponding numerical codes were theoretically verified at first. Then, a MIECT experimental system and a prototype probe were developed and the validity of the proposed numerical methods were further proved experimentally. It is found that the moving probe-fixed object method is the most efficient scheme for MIECT signal simulation among the three proposed numerical methods. The numerical methods and codes developed in this paper provide numerical tools for design and optimization of the MIECT probes and testing system used in inspection of rails or wheels system etc, as well as the quantitative defect evaluation based on the MIECT signals through model based inverse analyses.

Vector potential-based MHD solver for non-periodic flows using Fourier continuation expansions

A high-order method to evolve in time electromagnetic and velocity fields in conducting fluids with non-periodic boundaries is presented. The method has a small overhead compared with fast FFT-based pseudospectral methods in periodic domains. It uses the magnetic vector potential formulation for accurately enforcing the null divergence of the magnetic field, and allowing for different boundary conditions including perfectly conducting walls or vacuum surroundings, two cases relevant for many astrophysical, geophysical, and industrial flows. A spectral Fourier continuation method is used to accurately represent all fields and their spatial derivatives, allowing also for efficient solution of Poisson equations with different boundaries. A study of conducting flows at different Reynolds and Hartmann numbers, and with different boundary conditions, is presented to study convergence of the method and the accuracy of the solenoidal and boundary conditions.

Analytical modeling and experimental validation of the six pole axial active magnetic bearing

The article presents a novel design of the axial active magnetic bearing with six poles. An analytical magnetic bearing model was developed to provide the axial magnetic induction distribution in 3D. The model utilizes magnetic vector potential formulation and Schwarz-Christoffel mapping. The paper covers in-depth deliberations on the end effect influence and the conjugate complex permeance function. Numerical model and simulations were provided in 3D mode with support of COMSOL Multiphysics software. The six pole axial active magnetic bearing was manufactured and investigated experimentally. Identification of the magnetic field distribution was carried out with a single axis magnetic induction sensor using custom automatic field scanner. High convergence of the modeling results and experimental research was demonstrated. The main advantage of the proposed analytical method is significantly shorter computation time compared to the numerical one that is useful from the modeling and controller study point of view. The configuration, modeling, simulation and experimental investigation results are well illustrated for the better overview.

A review on methods to simulate three dimensional rotating electrical machine in magnetic vector potential formulation using edge finite element method under sliding surface principle

AbstractIn recent years edge finite element method has become quite popular in modeling the three‐dimensional rotating electrical machine due to its ability to represent the physical behavior of field along interface correctly. However, modeling becomes complex when significant mesh distortion occurs along a sliding surface due to movement. A remeshing step is needed, which increases computational cost. In order to avoid remeshing, efficient numerical methods have already been developed to model movement in the three‐dimensional rotating electrical machine under edge finite element method. Nevertheless, very little research has been carried out on comparing these methods. The main objective of this paper is to compare the edge‐based interpolation and mortar edge element method. To improve the accuracy of edge‐based interpolation for tetrahedral mesh, we propose an improved strategy. Numerical results illustrate the accuracy, robustness and performance of the methods. A detailed comparison is presented based on the criteria mentioned above.

Integration of Energetic Model for Ferromagnetic Hysteresis in Finite Volume Method for Electromagnetic Field Calculation

In this article, a coupled algorithm between the control volume method and the hysteresis dynamic energetic model for ferromagnetic hysteresis is presented. To illustrate the dynamic behavior of ferromagnetic materials, the quasi-static model is extended by adding two components to the applied magnetic field “Hedd”, and “Hexc”. The added fields are related to the excess losses and classical eddy losses. Thus, two new supplementary coefficients are added to the model parameters. The determination of those coefficients is attained by measuring the energy density for two distinct frequencies. This model introducing the magnetic induction as an independent variable is presented in order to be directly used in time-stepping finite volume calculations applied to the magnetic vector potential formulation. The calculated results are validated by experiences performed in an Epstein’s frame. To check the effectiveness of this model combined with the control volume method in the time domain, the obtained results are compared with experiments.

A Generic 3-D Analytical Model of Permanent Magnet Eddy-Current Couplings Using a Magnetic Vector Potential Formulation

In this article, a new and general three-dimensional (3-D) analytical frame for evaluating the performance characteristics of the permanent magnet (PM) eddy-current coupling is presented using a magnetic vector potential formulation. Based on the 3-D Cartesian coordinate system, the actual coupling geometry is equated to a linear structure by applying the mean radius (for cylindrical conductor and PM rotors) and the equivalent cuboid (for PMs of various shapes) assumptions. The torque contribution from the overhangs of conductor back iron and the effects of PM and air gap thicknesses are all taken into considered. A verification process is implemented and it is shown that, in any case, the proposed 3-D analytical model can yield the results very close to those obtained by both the 3-D finite-element analysis and experimental measurement.

Space-Time-Parallel 3-D Finite Element Transformer Model With Adaptive TLM and Parareal Techniques for Electromagnetic Transient Analysis

In this paper, a novel massively space-time parallel finite element (FE) transformer model is proposed for power system electromagnetic transient simulations. The method utilizes the reduced magnetic vector potential (RMVP) formulation to solve the electromagnetic field in the transformer, and the 3-D tetrahedral edge element is applied to discretize the RMVP formulation. The discretized nonlinear system is solved by adaptive transmission line modeling (TLM) and matrix-free conjugate gradient iterator, which allows parallelism at the elemental-level and eliminates the need for Newton-Raphson (NR) iterations over the large-scale global matrix. The paper also integrates the parallel-in-time method with the finite element solver to further parallelize the model at space-time level, and the GPU implementation of the model realized a speedup of over 55 and high accuracy, compared with a commercial FE software.

MODELING OFDISTRIBUTION TRANSFORMERS BY A 3-D NUMERICAL TECHNIQUE

Modeling of electromagnetic quantities (magnetic flux, leakage flux, eddy current loss, and electromagnetic force) in distributed power transformers remarkably plays an essential role for researchers and designers in the calculation and production of electrical equipment in general and transformer designs in particular. In the frame of this research, the distribution of magnetic flux densities along the tank wall and cover plate, the leakage flux in air regions, and the forces for both rated and short circuit modes in the windings of the transformer are computed/simulated via a 3-D numerical method. The method is herein developed with magnetic vector potential formulations and is applied to a practical application of the distributed transformer (630kVA-22/0,4kV).